Oriented Matroids: The Power of Unification

Matroids were invented by Hassler Whitney as a way of generalizing a variety of properties that were common to structures such as vector spaces and graphs (see a previous Feature Column, Matroids: The Value of Abstraction, for more information on matroids). The properties of bases of a vector space, all of which have the same size, are an example of a property of a vector space that is generalized in a matroid. One way of thinking of matroids is to have them developed in terms of the circuits in a graph. Digraphs (short for directed graphs), diagrams consisting of dots and line segments (edges) with arrows assigned to the edges, are a generalization of the graph concept. Analogously, oriented matroids are a generalization of matroids that mimic the way that matroids are related to graphs by looking at the circuit structure of digraphs. It is hard to single out the birth of the study of oriented matroids, but in a very short period of time matroids have proved to be a powerful way to view a variety of geometrical, combinatorial and topological ideas that have also pointed at a variety of applications. Early researchers in what emerged as the subject of oriented matroids were R. Bland, J. Folkman, M. Las Vergnas, and J. Lawrence. Despite the fact that they generalize a variety of geometrical phenomena, oriented matroids are associated with ideas that can be illustrated with concrete examples.

At one time Italy and Germany consisted of a lot of small provinces, each of which acted more or less independently of the other provinces in the area. This meant that there was no central system of planning, or currency, or standards, etc. When these provinces became unified into a single national state, much more could be accomplished. There is an analogy here for mathematics. When mathematical ideas are encapsulated into small chunks without the ties between the chunks, it is sometimes not possible to achieve the full power of what might be done. The concept of oriented matroids has helped unify a broad collection of seemingly isolated mathematical ideas (some arising in graph theory, the study of polytopes, and arrangements of lines) and that has encouraged researchers to find in other areas the analog of what has been fruitful in a specific area.